Drinfeld 双重上的模范畴的 Grothendieck 环
Grothendieck Rings of Module Categories over Drinfeld Doubles
AI总结：
本文研究 Drinfeld 双重上模范畴的 Grothendieck 环，将其实现为二次上循环装饰的双 Burnside 环，给出 Clifford 乘法公式，并刻画半单性、Brauer-Picard 作用的传递性及 Lagrangian 子群分类。
AI中文摘要：
设 $k$ 为特征零的代数闭域，$G$ 为有限群。我们将 ${\mathcal{R}\mathit{ep}}(D(G))$-模范畴的基于 Grothendieck 环 $R_G$ 实现为二次上循环装饰的双 Burnside 环，并推导出乘法和 $R_G$ 对 ${\operatorname{\operatorname{\mathsf{Vec}}}}_G$-模范畴的 Grothendieck 群作用的显式 Clifford 公式。我们确定了极值基于理想，研究了通过较小群的分解，并证明了标准双侧子群归纳的 Mackey 定理。我们证明 $\mathbb C\otimes_{\mathbb Z}R_G$ 当且仅当 $G$ 为循环群时是半单的。我们研究了 Brauer--Picard 作用在不可分解的 ${\operatorname{\operatorname{\mathsf{Vec}}}}_G$-模范畴上的性质，并证明该作用当且仅当 $G$ 为平方自由指数的交换群时是传递的。对于交换群 $G$，我们确定了 $G\oplus\widehat G$ 的 Lagrangian 子群的可能抽象群类型，在齐循环情形应用已知的正交分类，并展示了混合指数情形下同型非共轭的 Lagrangian。
英文摘要：
Let $k$ be an algebraically closed field of characteristic zero and $G$ a finite group. We realize the based Grothendieck ring $R_G$ of ${\mathcal{R}\mathit{ep}}(D(G))$-module categories as the degree-two cocycle-decorated double Burnside ring and derive an explicit Clifford formula for multiplication and for the action of $R_G$ on the Grothendieck group of ${\operatorname{\operatorname{\mathsf{Vec}}}}_G$-module categories. We determine the extremal based ideals, study factorization through smaller groups, and prove a Mackey theorem for standard two-sided subgroup inductions. We prove that $\mathbb C\otimes_{\mathbb Z}R_G$ is semisimple exactly when $G$ is cyclic. We study the Brauer--Picard action on indecomposable ${\operatorname{\operatorname{\mathsf{Vec}}}}_G$-module categories and show that it is transitive exactly when $G$ is abelian of square-free exponent. For abelian $G$, we determine the possible abstract group types of Lagrangian subgroups of $G\oplus\widehat G$, apply the known orthogonal classification in the homocyclic case, and exhibit same-type nonconjugate Lagrangians for mixed exponents.
